IntroductiontoQuantitativeEconomics
JesseM.Shapiro
TheMITPress
Cambridge,Massachusetts London,England
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Prefacevii
Introduction1
1Monopoly3
1.1RefutingtheModel3
1.2LearningfromtheModel5
2Entry13
2.1DatafromaSingleMarket13
2.2DatafromaSetofMarkets17
3Demand25
3.1DatafromTwoMarkets26
3.2DatafromThreeMarkets30
4Production37
4.1ASingleDecision37
4.2ManyDecisions41
5Information51
5.1DataonActionsandPayoffs54
5.2CounterfactualChanges56
5.3StrategicBehavior59
References65 Index67
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Preface
Thiscourseaimstointroducegraduatestudentstoquantitativeeconomics,bywhichImean theuseofeconomicmodelstointerpretdata.Thecourseassumesknowledgeofeconomic andeconometrictheoryatthelevelofanintroductorygraduatesequence.
Thecourseproceedsthroughaseriesofeconomicsettings.Eachsettingmotivatesa canonicaleconomicmodel.Studentsareaskedtoreflectonasetofquestions,toread proposedanswers,andthentoperformsomeexercises.Exercisesareacombinationof theoreticalexercisesdesignedtoextendconceptsandquantitativeexercisesdesignedto applythem.
Attheendofthecourse,studentsarepreparedtobeginusingeconomictheoryto interpretdata.Theyarealsopreparedtocontinuethestudyofquantitativeeconomicswithin particularsubfields,wheretheymayencountermoreelaborateorspecializedeconomic models,andmorecomplexdatastructures.
Thecoursereliesheavilyonstudentwork.Studentswhodonotpausetoconsider(and, ideally,writeanswersto)thereflectionquestions,orwhodonotworkthroughtheexercises, arelikelytogetmuchlessoutofthecoursethanstudentswhomakethoseinvestments.
ThemethodologicalperspectiveofthecoursebuildsonMarschak(1950).Ilearnedmany ofthespecificmodelsinthebookfrommyteachers,someofwhomhavewrittentextbooks (see,e.g.,Becker2007;Jaffeetal.2019).Formerteachingassistantsandstudentshavealso contributedtothedevelopmentofthecourse.
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Introduction
Quantitativeeconomicsisconcernedwiththeuseofeconomicmodelstointerpretdata.Its roleisillustratedinthefollowingstylizedworkflow.
Counterfactuals
Somemeasures ˆ Y areobtainedforsomeeconomicdata Y ∗ .Forexample,wemaymeasure incomes ˆ Y forasampleofindividualsinapopulationwithincomedistribution Y ∗ .An importanttaskofstatisticsistohelpmakeinferencesabouttheeconomicdata Y ∗ from themeasures ˆ Y ,forexamplebyusingamodelofthesamplingprocesstolearnaboutthe populationdistributionofincomefromthesampledistributionofincome.
Undersomeeconomicmodel,thedata Y ∗ canbeinformativeaboutsomeeconomic primitives .Forexample,knowledgeofthepopulationdistributionofincomemay informsomeprimitivesthatgoverntheincomeprocess.Animportanttaskofquantitative economicsistomakeinferencesabouttheprimitives fromthedata Y ∗ .Economictheory maythenbeusedtoderiveaparameterofinterest θ ( ),suchastheefficiencyofthe economy,oracounterfactualinstance θ oftheparameterunderalternativeprimitives, suchasanincometax.
Thiscoursewillbeespeciallyconcernedwithwhenandhowwecanlearnprimitives , parameters θ ( ),andcounterfactuals θ ofinterestfromeconomicdata Y ∗ .Thecourse willtouchoccasionallyonissuesofmeasurement,andonthemesadjacenttostatistics,but thesewillnotbefrontandcenter.
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Monopoly
Afirmsellsadistinctivegoodatsomeprice P∗ > 0toapopulationofcustomers.Thefirm sellsaquantity Q∗ = Q (P∗ ) > 0determinedbythestrictlydecreasinganddifferentiable demandfunction Q : R≥0 → R≥0 .Forsimplicity’ssake,wesupposethatthefirmhasa constantmarginalcost C > 0ofproduction.Anexampleofagoodsoldinthiswaymight beasubscriptionfromthelocalcablecompany,oranon-patentdrug.
Ourdataconsistof {P∗ , Q∗ },thatis,thepriceofthegoodandthequantitysold.We, theeconomist,positthatthefirmisamonopolistthatmaximizesstaticprofitsgivenby π (P) = Q (P)(P C ),possiblyuptoafixedcostthatdoesnotdependonthepriceor quantitysold.
Definition1.1. The staticmonopolymodel holdsthat
Itisapropertyofthestaticmonopolymodelthat P∗ > C .
1.1RefutingtheModel
Ifthemodelisinconsistentwiththedatathenwemaywanttorethinkthemodel.Therefore afirstquestionweshouldaskiswhetherorwhenthemodelcanberefuted.
Question1.1. Fromdata {P∗ , Q∗ },canweeverrefutethestaticmonopolymodel? Pausetoreflect.Writedownyouranswerbeforecontinuing. ©MIT Press. Not for redistribution.
Firstweshouldprobablysaywhatitmeansfordatatorefuteamodel.
Definition1.2. Considera model thatspecifiesaset Y ( ) ofadmissiblevaluesofsome observeddata Y givenprimitives .Agivenrealization Y ∗ ofthedata refutes themodelif therearenoprimitives suchthat Y ∗ ∈ Y ( ).
Inthesettingweareconsideringhere,thedataare Y ∗ = {P∗ , Q∗ },theprimitivesare = {Q (·) , C },andthemodelisthestaticmonopolymodelthatspecifiesasetoffeasiblepairs {P, Q} foranyprimitives . Fromdataonmarketpriceandquantityalone,wecannotrefutethestaticmonopoly model.Exercise1.1verifiesthisformally.Intuitively,sincewehaveputveryfewrestrictions onthemodel,itisalwayspossibletofindsomeprimitivestojustifyanyprice-quantitypair. Itmayhelptoenrichwhatwecanmeasure.Whatifwewecouldobservethedemand function Q (·)?
Question1.2. Fromdata {P∗ , Q (·)},canweeverrefutethestaticmonopolymodel?
Pausetoreflect.Writedownyouranswerbeforecontinuing.
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Itmayseemstrangetothinkofafunction Q (·) aspartofourdata,andindeeditis abitstrange,sincewewillrarelyifevermeasuresucharichobjectdirectlyinthereal world.Fornowwecanviewitasakindofbookendforwhatwecouldhopetolearn aboutthedemandfunction.Ifwecan’trefutethemodelevenwithfullknowledgeof Q (·),thenwecan’trefuteitwiththeincompleteinformationabout Q (·) thatwecanmore realisticallyexpect.If,ontheotherhand,fullknowledgeof Q (·) doesallowustorefute themodel,thenunderstandinghowmaypointthewaytowardrelaxingtherequirementsfor thedata.
And,indeed,themodelcanberefutedinthiscase.Toseehow,recallthatanecessary conditionforanoptimalprice P∗ > 0isthat
whichcanbewrittenas
where Q∗ = Q (P∗ ) and ε (P∗ ) < 0isthepriceelasticityofdemandevaluatedatprice P∗ ThisisthefamousLernerRuleforoptimalmonopolypricing. Because P∗ > C > 0itisimmediatethat ε (P∗ ) < 1,orthatthemonopolistpricesonthe elasticpartofthedemandcurve.Thereforeif Q ( ) issuchthat 1 ≤ ε (P∗ ) < 0,thestatic monopolymodelisrefuted.Itfollowsthat,tohavetheprospectofrefutingthemodel,itis notnecessarytomeasuretheentiredemandfunction Q ( ).Measuringitsderivativelocalto theobservedprice P∗ issufficient.
Fact1.1. Fromdata P∗ , Q∗ , Q (P∗ ) itispossibletorefutethestaticmonopolymodel,in particularif Q (P∗ ) ≤ Q∗ P∗ .
Fact1.1isusefulinevaluatingclaimsofmonopolization.Forexample,typicalestimates indicatethatworldoildemandisinelasticeveninthelongrun(Smith,2009).Thisrefutes thehypothesisthattheworldoilmarketismonopolized,forexamplebyOPEC.Fact1.1 holdsundermoregeneralformsofthemonopolist’scostfunction.Exercise1.2showsthis. Fact1.1alsoillustratesamoregeneralprincipleofquantitativeeconomics.
Principle1. Economicmodelsimposerestrictionsoneconomicdata,andthese restrictionsareusefulindistinguishingamongdifferentmodels.
1.2LearningfromtheModel
Monopolycanleadtoeconomicinefficiencybecausepricingagoodaboveitsmarginalcost precludessometransactionsthatwouldbenefitboththebuyerandtheseller.Beginningat ©MIT Press. Not for redistribution.
themonopolyquantity Q∗ ,thegaininsurplusfromanincremental(infinitesimal)increase inthequantityisgivenby
= P∗ C ,(1.2)
whichisalsothemonopolist’sper-unitprofit.Intuitively, P∗ representsthevaluetothebuyer ofanincrementalsaleand C representsitscosttotheseller.2
Knowingthevalueof λ maybeuseful,forexampletoevaluatethepotentialbenefit fromquantityregulationorcompetitionpolicy.Itisobviousthatknowledgeof P∗ and C is sufficienttolearn λ.Insomemarketsitmaybeatallordertoasktheeconomisttomeasure themonopolist’smarginalcostdirectly.Canwedowithoutit?
Question1.3. Fromdata {P∗ , Q∗ },canweidentify λ underthestaticmonopoly model?
Pausetoreflect.Writedownyouranswerbeforecontinuing.
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Firstweshouldprobablysaywhatitmeanstoidentifyaneconomicparameter.
Definition1.3. Consideramodelthat,givenprimitives ,specifiesaset Y ( ) of admissiblevaluesofsomeobserveddata,andavalue θ ( ) ofsomeparameter.The parameter θ is identified if Y ∗ ∈ Y ( ), Y =⇒ θ ( ) = θ .Thatis,theparameter isidentifiedifprimitivesthatcanyieldidenticaldataalwaysimplyidenticalvaluesofthe parameter.
Inthesettingweareconsideringhere,thedataare Y ∗ = {P∗ , Q∗ },theprimitivesare = {Q (·) , C },theparameteristheincrementalgaininsurplus λ,andthemodelisthestatic monopolymodel.
Fromdataonmarketpriceandquantityalone,wecannotidentifytheincrementalgainin surplus λ.(Exercise1.3provesastrongerformofthisstatement.)Whatifwecouldmeasure somethingmoreaboutthedemandfunction?
Question1.4. Fromdata P∗ , Q∗ , Q (P∗ ) ,canweidentify λ underthestatic monopolymodel?
Pausetoreflect.Writedownyouranswerbeforecontinuing.
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Thereasoningbehindfact1.1ishelpfulhere.Fromdata P∗ , Q∗ , Q (P∗ ) ,weknowthe elasticity ε (P∗ ) ofdemandattheequilibriumprice P∗ .Fromequation(1.1),wetherefore knowtheLernerindex (P∗ C ) /P∗ ,andhenceweknowthecost C fromtheprice P∗ .It followsthattheanswertoquestion1.4isyes.
Fact1.2. Underthestaticmonopolymodel,thecost C ,andthereforetheincrementalgain insurplus λ,isidentifiedfromdata P∗ , Q∗ , Q (P∗ )
Fact1.2isusefulbecauseitallowstheeconomisttolearntheincrementalsurplus λ withoutdirectinformationonthemonopolist’scosts.Forexample,ifweknowthedemand Q (·) foradrugsoldbyamonopolypatent-holder,wecanassesstheincrementalsurplus fromaregulatedincreaseinquantitysoldwithoutdirectinformationonthecostsof manufacturinganddistribution.Exercise1.4showsthatastatementsimilartofact1.2holds fortheincrementalgaininsurplusfromareductioninprice.
Fact1.2alsoillustratesamoregeneralprincipleofquantitativeeconomics.
Principle2. Economicmodelstietogetherdifferenteconomicobjects,andtheseties canmakeiteasiertolearnwhatwewanttoknow.
Principle2isimportantinpartbecauseoftenwhatweseektoknowisnotdirectly observable.
Definition1.4. Consideramodelthat,givenprimitives ,specifiesavalue θ ( ) ofsome economicobject.Consideranalternativemodelthat,givenprimitives ˜ ,specifiesavalue ˜ θ ˜ ofthesameobject.Thenthe counterfactual changein θ ,givenby ˜ θ ˜ θ ( ),is thechangeinthevalueof θ broughtaboutbychangingtheprimitivesfrom to andthe modelfrom θ (·) to θ (·).
Forexample,equation(1.2)givesthecounterfactualchangeintotalsurplusifweholdthe primitives = {Q ( ) , C } constantbutmodifythemodeltosetthequantityincrementally abovetheonesetbythestaticmonopolist.
Exercises
Exercise1.1. Thereexistsnorealization {P∗ , Q∗ } ∈ R2 ≥0 thatrefutesthestaticmonopoly model.Provethis. ©MIT Press. Not for redistribution.
Definition1.5. The generalizedstaticmonopolymodel holdsthat
∗
(P)
(P) = Q (P) P C (Q (P))
∗ = Q P∗ for C : R≥0 → R≥0 ,adifferentiableandstrictlyincreasingtotalcostfunction.
Exercise1.2. Fact1.1holdsunderthegeneralizedstaticmonopolymodel.Provethis.
Exercise1.3. Underthestaticmonopolymodel,theincrementalgaininsurplus λ is notidentifiedfromdata {P∗ , Q∗ }.Inparticular,foranysuchdatawith P∗ , Q∗ > 0,any incrementalgaininsurplus0 <λ< P∗ isconsistentwiththestaticmonopolymodel.Prove this.
Exercise1.4. Underthestaticmonopolymodel,ananalogueoffact1.2holdsforthe incrementalgaininsurplusfromareductionintheprice P∗ .Provethis.
Exercise1.5. Table1.1showsdataontheglobalpriceandquantityofvitaminCfortwo yearsinthe1990s.Aneconomistpositsthemodel Qt = Q (Pt ) for Q (·) astrictlydecreasing anddifferentiabledemandfunction.Istheeconomist’smodelrefutedbythedatain table1.1?
Table1.1
GlobalpriceandquantityofvitaminC,1990and1994
Source: IgamiandSugaya(2021)
Exercise1.6. Theeconomistinexercise1.5revisesthemodeltoholdthat Qt = expα +β t Pt , where α , β ,and arescalarparameters, < 0,and t isthecalendaryear.Istherevisedmodel refutedbythedataintable1.1?
Exercise1.7. Underthemodelinexercise1.6,istheparameter θ = (α , β , ) identifiedfrom data Y = {(Pt , Qt )}t =t ,t fordistinctyears t = t ?
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Exercise1.8. Nowsupposethedata Y = {(Pt , Qt )}t =t ,t ,t areforthreedistinctyears t < t < t .Giveanexampleofconditionsonthedataunderwhichthemodelinexercise1.6 isrefuted.Istheparameter θ = (α , β , ) identified?
Exercise1.9. Table1.2addsanadditionalyearofdatarelativetotable1.1.Giventhemodel inexercise1.6,infertheparameter fromthedataintable1.2.
Table1.2
GlobalpriceandquantityofvitaminC,1989,1990,and1994
Source: IgamiandSugaya(2021)
Exercise1.10. In1994asignificantportionofthemarketforvitaminCwascontrolledby acartel(Connor,2008;IgamiandSugaya,2022).Supposethatin1994thecartelcontrolled theentiremarketandactedaccordingtothestaticmonopolymodel,withademandfunction givenbythesolutiontoexercise1.9.Whatwasthecartel’scost C1994 forproducinga kilogramofvitaminC?
Exercise1.11. Undertheconditionsofexercise1.10,whatwouldhavebeenthegainin surplusfromincrementallyincreasingthequantityofvitaminCin1994?
Exercise1.12. Undertheconditionsofexercise1.10,whatwouldhavebeenthelossin profittothecartelfromreducingthepriceofvitaminCin1994toits1990price,leaving thecostofproduction,andthedemandfunction,unchanged?
Exercise1.13. Undertheconditionsofexercise1.10,whatwouldhavebeenthegainin consumersurplusfromreducingthepriceofvitaminCin1994toits1990price,leaving thedemandfunctionunchanged?
Fortheremainingexerciseswe’llintroduceanewmodel.Inthismodel,adominant firmchoosesaprice P∗ > 0.Ratherthansellingthefullquantitydemanded Q∗ = Q (P∗ ) > 0,however,thedominantfirmsellsanamountgivenby Q∗ F ∗ ,where F ∗ = F (P) is anamountsuppliedbyacompetitivefringeofprice-takingproducers,whoseoutputis determinedbythestrictlyincreasinganddifferentiablesupplyfunction F : R≥0 → R≥0 Asinthemonopolymodel,wecontinuetosupposethatthedominantfirmhasaconstant marginalcost C > 0ofproduction.
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Definition1.6. The dominantfirmmodel holdsthat
Exercise1.14. Supposethatthedominantfirmmodelholdsandthat0 < F ∗ < Q∗ .Derive andinterpretthefirst-orderconditionforoptimalpricingbythedominantfirm.
Exercise1.15. Asthecartelgrew,sodidthesupplyfromacompetitivefringeexcluded fromthecartel’sarrangements.Table1.3includesinformationontheamountofvitaminC suppliedbythiscompetitivefringe.Supposethat F (P) = F0 Pφ t for F0 > 0aconstant.What istheelasticity φ> 0?
Table1.3
Globalprice,totalquantity,andfringequantityofvitaminC,1989, 1990,and1994
Source: IgamiandSugaya(2021)
Exercise1.16. Supposethat,in1994,thedominantfirmmodelheld,withthecartelinthe roleofthedominantfirm,thedemandfunctiongiveninexercise1.6,andthefringesupply functiongiveninexercise1.15.Whatwasthecartel’scost C1994 forproducingakilogram ofvitaminC?
Exercise1.17. Answerthequestionsinexercises1.12and1.13undertheconditionsof exercise1.16.
Exercise1.18. Undertheconditionsofexercise1.16,whatwouldhavebeenthegainin totalsurplusfromreplacingallfringeproductionwithcartelproductionin1994,leaving thetotalquantitysoldunchanged?
Notes
1.Ifnot,then π P∗ > 0,acontradiction.
2.Thetotaldeadweightlossfrommonopolyisgivenby Q(C ) Q∗ (P (Q) C ) dQ for P (Q) theinversedemand functionwith Q (P (Q)) = Q.Theexpressioninequation(1.2)thenfollowsbythefundamentaltheoremofcalculus. ©MIT Press. Not for redistribution.